| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | Specimen |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question on polar coordinates requiring sketching a cardioid, computing area using the polar area formula, deriving an arc length result involving the identity r² + (dr/dθ)² and double-angle formulas, then integrating to find arc length. While the techniques are standard for FM students, the arc length derivation requires algebraic manipulation and the integration needs a substitution or half-angle knowledge, making it moderately challenging but within expected FM scope. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The curve $C$ has polar equation $r = a ( 1 - \cos \theta )$ for $0 \leqslant \theta < 2 \pi$.\\
(i) Sketch $C$.\\
(ii) Find the area of the region enclosed by the arc of $C$ for which $\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$, the half-line $\theta = \frac { 1 } { 2 } \pi$ and the half-line $\theta = \frac { 3 } { 2 } \pi$.\\
(iii) Show that
$$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$
where $s$ denotes arc length, and find the length of the arc of $C$ for which $\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q11 OR}}